Theory of Statistics

718 Pages·1995·30.593 MB·English

by  Mark J. Schervish (auth.)

Springer Series in Statistics
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P. Diggie, S. Fienberg, K. Krickeberg,
1. Oikin, N. Wermuth
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AndersenlBorganlGilllKeiding: Statistical Models Based on Counting Processes.
Andrews/Herzberg: Data: A Collection of Problems from Many Fields for the Student
and Research Worker.
Anscombe: Computing in Statistical Science through APL.
Berger: Statistical Decision Theory and Bayesian Analysis, 2nd edition.
Bolfarine/Zacks: Prediction Theory for Finite Populations.
Borg/Groenen: Modem Multidimensional Scaling: Theory and Applications
Bremaud: Point Processes and Queues: Martingale Dynamics.
BrockwelllDavis: Time Series: Theory and Methods, 2nd edition.
Daley/Vere-Jones: An Introduction to the Theory of Point Processes.
Dzhaparidze: Parameter Estimation and Hypothesis Testing in Spectral Analysis of
Stationary Time Series.
Fahrmeir/Tutz: Multivariate Statistical Modelling Based on Generalized Linear
Models.
Farrell: Multivariate Calculation.
Federer: Statistical Design and Analysis for Intercropping Experiments.
Fienberg/HoaglinlKruskal/Tanur (Eds.): A Statistical Model: Frederick Mosteller's
Contributions to Statistics, Science and Public Policy.
Fisher/Sen: The Collected Works of Wassily Hoeffding.
Good: Pennutation Tests: A Practical Guide to Resampling Methods for Testing
Hypotheses.
GoodmanlKruskal: Measures of Association for Cross Classifications.
Grandell: Aspects of Risk Theory.
Haberman: Advanced Statistics, Volume I: Description of Populations.
Hall: The Bootstrap and Edgeworth Expansion.
Hardie: Smoothing Techniques: With Implementation in S.
Hartigan: Bayes Theory.
Heyer: Theory of Statistical Experiments.
Huet/Bouvier/GruetlJolivet: Statistical Tools for Nonlinear Regression: A Practical
Guide with S-PLUS Examples.
Jolliffe: Principal Component Analysis.
KolenlBrennan: Test Equating: Methods and Practices.
Kotz/Johnson (Eds.): Breakthroughs in Statistics Volume I.
Kotz/Johnson (Eds.): Breakthroughs in Statistics Volume II.
Kres: Statistical Tables for Multivariate Analysis.
Le Cam: Asymptotic Methods in Statistical Decision Theory.
Le Cam/Yang: Asymptotics in Statistics: Some Basic Concepts.
Longford: Models for Uncertainty in Educational Testing.
Manoukian: Modem Concepts and Theorems of Mathematical Statistics.
Miller, Jr.: Simultaneous Statistical Inference, 2nd edition.
Mosteller/Wallace: Applied Bayesian and Classical Inference: The Case of The
Federalist Papers.
tc onlin"ltd after indu)
Mark J. Schervish
Theory of Statistics
With 26 Illustrations
t
Springer
Mark J. Schervish
Department of Statistics
Carnegie Mellon University
Pittsburgh, PA 15213
USA
Library of Congress Cataloging-in. Publication Data
Schervish, Mark J.
Theory of Statistics / Mark J. Schervish
p. cm. - (Springer series in statistics)
Includes bibliographical references (po ) and index.
ISBN· 13: 978-1-4612-8708-7
1. Mathematical statistics. L Title. II. Series.
QA276.S346 1995
519.5--dc20 95-11235
Printed on acid-free paper.
© 1995 Springer-Verlag New York, Inc.
Softcover reprint of the hardcover 1ste dition 1995
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Production managed by Laura Carlson; manufacturing supervised by Joe Quatela.
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Printed in the United States of America.
9 8 7 6 5 4 3(C 2o lTe(:ted second printing. 1997)
ISBN-13: 978·1-4612-8708-7 e-ISBN·13: 978-1-4612-4250-5
DOl: 10.10071978-1-4612-4250-5
To Nancy, Margaret, and Meredith
Preface
This text has grown out of notes used for lectures in a course entitled Ad
vanced Statistical Theory at Carnegie Mellon University over several years.
The course (when taught by the author) has attempted to cover, in one
academic year, those topics in estimation, testing, and large sample theory
that are commonly taught to second year graduate students in a math
ematically rigorous fashion. Most texts at this level fall into one of two
categories. They either ignore the Bayesian point of view altogether or
they cover Bayesian topics almost exclusively. This book covers topics in
both classicaJl and Bayesian inference in a great deal of generality. My own
point of view is Bayesian, but I believe that students need to learn both
types of theory in order to achieve a fuller appreciation of the subject mat
ter. Although many comparisons are made between classical and Bayesian
methods, it is not a goal of the text to present a formal comparison of the
two approaches as was done by Barnett (1982). Rather, the goal has been
to prepare Ph.D. students to be able to understand and contribute to the
literature of theoretical statistics with a broader perspective than would be
achieved from a purely Bayesian or a purely classical course.
After a brief review of elementary statistical theory, the coverage of the
subject matter begins with a detailed treatment of parametric statistical
models as motivated by DeFinetti's representation theorem for exchangeable
random variables (Cha{>ter 1). In addition, Dirichlet processes and other
tailfree processes are presented as examples of infinite-dimensional param
eters. Chapter 2 introduces sufficient statistics from both Bayesian and
non-Bayesian viewpoints. Exponential families are discussed here because
of the important role sufficiency plays in these models. Also, the concept
of information is introduced together with its relationship to sufficiency.
A representation theorem is given for general distributions based on suffi
cient statistics. Decision theory is the subject of Chapter 3, which includes
discussions of admissibility and minimaxity. Section 3.3 presents an ax
iomatic derivation of Bayesian decision theory, including the use of condi
tional probability. Chapter 4 covers hypothesis testing, including unbiased
tests, P-values, and Bayes factors. We highlight the contrasts between the
traditional "uniformly most powerful" (UMP) approach to testing and de
cision theoretic approaches (both Bayesian and classical). In particular, we
1 What I call classical inference is called frequentist inference by some other
authors.
viii Preface
see how the asymmetric treatment of hypotheses and alternatives in the
UMP approach accounts for much of the difference. Point and set estima
tion are the topics of Chapter 5. This includes unbiased and maximum like
lihood estimation as well as confidence, prediction, and tolerance sets. We
also introduce robust estimation and the bootstrap. Equivariant decision
rules are covered in Chapter 6. In Section 6.2.2, we debunk the common
misconception of equivariant rules as means for preserving decisions un
der changes of measurement scale. Large sample theory is the subject of
Chapter 7. This includes asymptotic properties of sample quantiles, maxi
mum likelihood estimators, robust estimators, and posterior distributions.
The last two chapters cover situations in which the random variables are
not modeled as being exchangeable. Hierarchical models (Chapter 8) are
useful for data arrays. Here, the parameters of the model can be modeled
as exchangeable while the observables are only partially exchangeable. We
introduce the popular computational tool known as Markov chain Monte
Carlo, Gibbs sampling, or successive substitution sampling, which is very
useful for fitting hierarchical models. Some topics in sequential analysis are
presented in Chapter 9. These include classical tests, Bayesilill decisions,
confidence sets, and the issue of sampling to a foregone conclusion.
The presentation of material is intended to be very general and very pre
cise. One of the goals of this book was to be the place where the proofs could
be found for many of those theorems whose proofs were "beyond the scope
of the course" in elementary or intermediate courses. For this reason, it is
useful to rely on measure theoretic probability. Since many students have·
not studied measure theory and probability recently or at all, I have in
cluded appendices on measure theory (Appendix A) and probability theory
(Appendix B).2 Even those who have measure theory in their background
can benefit from seeing these topics discussed briefly and working through
some problems. At the beginnings of these two appendices, I have given
overviews of the important definitions and results. These should serve as
reminders for those who already know the material and as groundbreaking
for those who do not. There are, however, some topics covered in Ap
pendix B that are not part of traditional probability courses. In particular,
there is the material in Section B.3.3 on conditional densities with respect
to nonproduct measures. Also, there is Section B.6, which attempts to use
the ideas of gambling to motivate the mathematical definition of proba
bility. Since conditional independence and the law of total probability are
so central to Bayesian predictive inference, readers may want to study the
material in Sections B.3.4 and B.3.5 also.
Appendix C lists purely mathematical theorems that are used in the text
2These two appendices contain sufficient detail to serve as the. basis for.a ful~­
semester (or more) course in measure and probability. They are mcluded m thiS
book to make it more self-contained for students who do not have a background
in measure theory.
Preface ix
without proof, and Appendix D gives a brief summary of the distributions
that are used throughout the text. An index is provided for notation and
abbreviations that are used at a considerable distance from where they are
defined. Throughout the book, I have added footnotes to those results that
are of interest mainly through their value in proving other results. These
footnotes indicate where the results are used explicitly elsewhere in the
book. This is intended as an aid to instructors who wish to select which
results to prove in detail and which to mention only in passing. A single
numbering system is used within each chapter and includes theorems, lem
mas, definitions, corollaries, propositions, assumptions, examples, tables,
figures, and equations in order to make them easier to locate when needed.
I was reluctant to mark sections to indicate which ones could be skipped
without interrupting the flow of the text because I was afraid that readers
would interpret such markings as signs that the material was not impor
tant. However, because there may be too much material to cover, especially
if the measure theory and probability appendices are covered, I have de
cided to mark two different kinds of sections whose material is used at most
sparingly in other parts of the text. Those sections marked with a plus sign
(+ ) make use of the theory of martingales. A lot of the material in some
of these sections is used in other such sections, but the remainder of the
text is relatively free of martingales. Martingales are particularly useful in
proving limit theorems for conditional probabilities. The remaining sections
that can be skipped or covered out of order without seriously interrupting
the flow of material are marked with an asterisk (*). No such system is
foolproof, however. For example, even though essentially all of the material
dealing with equivariance is isolated in Chapter 6, there is one example in
Chapter 7 and one exercise that make reference to the material. Similarly,
the material from other sections marked with the asterisk may occasion
ally appear in examples later in the text. But these occurrences should be
inconsequential. Of course, any instructor who feels that equivariance is an
important topic should not be put off by the asterisk. In that same vein,
students really ought to be made aware of what the main theorems in Sec
tion 3.3 say (Theorems 3.108 and 3.110), even though the section could be
skipped without interrupting the flow of the material.
I would like to thank many people who helped me to write this book or
who read early drafts. Many people have provided corrections and guidance
for clarifying some of the discussions (not to mention corrections to some
proofs). In particular, thanks are due to Chris Andrews, Bogdan Doytchi
nov, Petros Hadjicostas, Tao Jiang, Rob Kass, Agostino Nobile, Shingo
Oue, and Thomas Short. Morris DeGroot helped me to understand what
is really going on with equivariance. Teddy Seidenfeld introduced me to
the axiomatic foundations of decision theory. Mel Novick introduced me
to the writings of DeFinetti. Persi Diaconis and Bill Strawderman made
valuable suggestions after reading drafts of the book, and those suggestions
are incorporated here. Special thanks go to Larry Wasserman, who taught
x
from two early drafts of the text and provided invaluable feedback on the
(lack of) clarity in various sections.
As a student at the University of Illinois at Urbana-Champaign, I learned
statistical theory from Stephen Portnoy, Robert Wijsman, and Robert
Bohrer (although some of these people may deny that fact after reading this
book). Many of the proofs and results in this text bear startling resemblance
to my notes taken as a student. Many, in turn, undoubtedly resemble works
recorded in other places. Whenever I have essentially lifted, or cosmetically
modified, or even only been deeply inspired by a published source, I have
cited that source in the text. If results copied from my notes as a student
or produced independently also resemble published results, I can only apol
ogize for not having taken enough time to seek out the earliest published
reference for every result and proof in the text. Similarly, the problems at
the ends of each chapter have come from many sources. One source used
often was the file of old qualifying exams from the Department of Statistics
at Carnegie Mellon University. These problems, in turn, came from various
sources unknown to me (even the ones I wrote). If I have used a problem
without giving proper credit, please take it as a compliment. Some of the
more challenging problems have been identified with an asterisk (*) after
the problem number. Many of the plots in the text were produced using
The New S Language and S-Plus [see Becker, Chambers, and Wilks (1988)
and StatSci (1992)]. The original text processing was done using U.TEJX,
which was written by Lamport (1986) and was based on 'lEX by Knuth
(1984).
Pittsburgh, Pennsylvania MARK J. SCHERVISH
May 91,1995
Several corrections needed to be made between the first and second print
ings of this book. During that time, I created a world-wide web page
http://www.stat.cmu.edu/-mark/advt/
on which readers may find up-~o-date lists of any corrections that have
been required. The most significant individual corrections made between
the first and second printings are listed here:
• The discussion of the famous M-estimator on page 314 has been
corrected.
• Theorems 7.108 and 7.116 each needed an additional condition con
cerning uniform bounded ness of the derivatives of the Hn and H;
functions on a compact set. Only small changes were made to the
proofs.
• The proofs of Theorems B.83 and B.133 were corrected, and small
changes were made to Example 2.81 and Definition B.137.
Contents
vii
Preface
Chapter 1: Probability Models 1
1.1 Background........ 1
1.1.1 General Concepts . 1
1.1.2 Classical Statistics 2
1.1.3 Bayesian Statistics 4
1.2 Exchangeability...... 5
1.2.1 Distributional Symmetry 5
1.2.2 Frequency and Exchangeability 10
1.3 Parametric Models . . . . . . . . . . . 12
1.3.1 Prior, Posterior, and Predictive Distributions 13
1.3.2 Improper Prior Distributions . . . 19
1.3.3 Choosing Probability Distributions 21
1.4 DeFinetti's Representation Theorem . 24
1.4.1 Understanding the Theorems . 24
1.4.2 The Mathematical Statements 26
1.4.3 Some Examples . . . . . . . . . 28
1.5 Proofs of DeFinetti's Theorem and Related Results' 33
1.5.1 Strong Law of Large Numbers 33
1.5.2 The Bernoulli Case . . . . 36
1.5.3 The General Finite Case' . . . 38
1.5.4 The General Infinite Case . . . 45
1.5.5 Formal Introduction to Parametric Models' 49
1.6 Infinite-Dimensional Parameters' 52
1.6.1 Dirichlet Processes 52
1.6.2 Tailfree Processes+ 60
1. 7 Problems ........ . 73
Chapter 2: Sufficient Statistics 82
2.1 Definitions ......... . 82
2.1.1 Notational Overview 82
2.1.2 Sufficiency ..... . 83
2.1.3 Minimal and Complete Sufficiency 92
2.1.4 Ancillarity............ 95
2.2 Exponential Families of Distributions. . . .102
'Sections and chapters marked with an asterisk may be skipped or covered
out of order without interrupting the flow of ideas.
+Sections marked with a plus sign include results which rely on the theory of
martingales. They may be skipped without interrupting the flow of ideas.